Download Geometric Sequences

Mathematical Investigations II
Name:
Functions – Teil Drei
The Exponential
Geometric Sequences
1.
A quick review: Which of the following are arithmetic sequences?
a.
5, 8, 11, 14, ...
b.
17, 13, 9, 5, 1, ...
c.
1, 4, 9, 16, ...
d.
1, 3, 5, 7, 9, ...
e.
4, 6, 9, 13.5, ...
f.
17
13
, 5,
, ...
3
3
2.
What was the basis for your decisions?
3.
Write both the recursive and explicit functions describing an arithmetic sequence with the
first term being 'a' and the common difference being 'd':


an  


if n  1
if n  1
a(n) =
4.
What common characteristic(s) do the following sequences have?
a.
1, 5, 25, 125, 625, 3125, ...
b.
8, 4, 2, 1, 0.5, 0.25, ...
c.
18, 12, 8,
16
, ...
3
d.
4, -12, 36, –108, ...

Exp. 8.1
Rev. S05
Mathematical Investigations II
Name:
5.
6.
Write a recursive function for each sequence in problem 4.
a.


an  


if n  1
c.


an  


if n  1
if n  1
if n  1
b.


an  


if n  1
d.


an  


if n  1
if n  1
if n  1
Write an explicit function for each sequence in part 4.
a.
a(n) =
b.
a(n) =
c.
a(n) =
d.
a(n) =
The sequences in part 4 are called geometric sequences. As you may have already noted, the
characteristic of a geometric sequence is that sequential terms have a common ratio; or
an
r
an1
where r = any non-zero real number and r  1, n  0 .
For the examples in part 4, (a) r = 5, (b) r =
7.
1
2
, (c) r = , (d) r = –3.
2
3
If a is the first term in geometric sequence and r is the common ratio,
a.
write out the first four terms of the sequence:
b.
write out a recursive function describing the geometric sequence:


an  


c.
if n  1
if n  1
write out an explicit function describing the geometric sequence:
a(n) =
Exp. 8.2
Rev. S05
Mathematical Investigations II
Name:
Some sample problems:
8.
If the first term of a geometric
sequence is 5 and the common ratio is
3, what is the value of the 5th term?
10.
If the third term of a geometric
11.
th
sequence. is 12 and the 7 term is .75,
what is the common ratio? (Be careful,
there are two possible answers.)
12.
Consider the sequence of squares
formed by connecting the midpoints of a.
the sides of a square whose sides are
10 units long.
d.
9.
If the first term of a geometric
sequence is 27 and the fourth term is
125, what is the common ratio?
Find the first and fourth terms of a
geometric sequence. with a5   0.2
and a8  0.4 .
Find the length of C5D5.
b.
Find a formula for the length of AnBn ,
where n > 1.
c.
Find a formula for the perimeter of the
nth square.
Find the area of A1B1C1D1.
Find the area of A3B3C3D3.
e.
Exp. 8.3
Find the area of AnBnCnDn .
Rev. S05
Mathematical Investigations II
Name:
The geometric mean of two positive numbers, a and b, is a third positive number, m, such that a,
m and b form a geometric sequence.
13.
14.
Find the geometric mean of
a.
4 and 9
b.
75 and 48
Find an expression for m in terms of a and b:
m=
15.
if 6 is the geometric mean of 3 and x, find x.
16.
Given right triangle ABC with AD an altitude,
a.
if CD = 3, and DB = 12, find the length
of AD. [Hint: Look for similar
triangles.]
b.
prove that AD is the geometric mean of DB and DC.
Exp. 8.4
Rev. S05